Resumen de The $L^1$-contraction principle in optimal transport
Matt Jacobs, Inwon C. Kim, Jiajun Tong
In this work we use the JKO scheme to approximate a general class of diffusion problems generated by Darcy’s law. Although the scheme is now classical, if the energy density is spatially inhomogeneous or irregular, many standard methods cannot be applied to establish convergence to the continuum limit. To overcome these difficulties, we analyze the scheme through its dual problem and establish a novel L1-contraction principle for the density variable. As a consequence, we obtain a new L1-equicontinuity property for the discrete solutions, which gives rise to strong L1-convergence of the densities to a weak solution of the continuum equation. Notably, the contraction principle relies only on the existence of an optimal transport map and the convexity structure of the energy. As a result, the principle holds in a very general setting, and opens the door to using optimal-transport-based variational schemes to study a larger class of non-linear inhomogeneous parabolic equations.
